2 Answers
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While most commentaries (Rosh, Rashbam, R'Yona) I have seen do not want to translate וְיִשְׁאֲלוּ as borrowing, I saw that the Chizkuni does. The reason he gives is so that the Mitzriim will subsequently run after the Jews.

Combine that with the Seforno who says that Moshe had to use diplomacy when telling the people so they shouldn't worry about the Mitzriim chasing them, for that will be their salvation!

The Shaul Umayshiv in his Commentary on the Torah Divrie Shaul gives a Halachic answer. They had to Borrow because the Shulchan aruch says that gold and silver of Idol Worshipers has a law like an Idol and it is Unlawfill to own it, there is another Law that a Jew can not Nullify an Idol. Hence they borrowed the items in order that after the Plague of the firstborn the gemara in Avodah Zarah says they said about their own Idols"They are worthless, as they where unable to save us from this terrible plague" hence nullifying their Idols. Therfore the Jews can now keep the gold, if they would have legally acquired it by taking it as gifts earlier, they would not have been unable to keep it as it was an Idol now that they borrowed it and the Egyptians Nullified it it is theirs for the Keeping. This answers the other question too that is to ask people to ask to borrow is uncomfortable as you dont want to seem like you are begging, but to take from Hefker like after the Splitting of the sea you see the opposite happened and Moshe had to tell them to leave.